perhaps the most studied varieties in algebraic geometry are the moduli spaces of smooth and stable curves. a stable curve is a nodal curve whose canonical divisor is ample. adding stable curves gives a geometrically meaning compactification to the space of smooth curves. a motivating problem in higher dimensional geometry is to construct the moduli space of varieties of general type, in any dimension. just as with curves, we need to consider non-normal varieties, possibly with more than one component. unlike the case of curves, if we fix the degree, it is not at all clear how to bound the number of components.